How to cite the article:
Libertini G. Concordance of the Predictions of a Simulation Model for the Evolutionary Advantage of Sex with Observational Evidence. WebmedCentral ZOOLOGY 2011;2(11):WMC002464
doi:
10.9754/journal.wmc.2011.002464

Abstract

Evolutionary advantage of sex is a widely discussed topic with multiple and clashing theories. A simulation model is proposed, based on the “classic” hypothesis that sex is advantageous because it allows faster attainment of favourable genetic combinations.The model shows the substitution of 2 (or 3) genes with advantageous alleles and calculates in which conditions a further gene allowing recombination is advantaged or disadvantaged in comparison with an allele not allowing recombination. With no epistasis, in infinite population sex results neutral, while in finite populations, in particular if the population is divided in demes, sex results advantageous.Considering the disadvantages caused by mating necessities, “classic” theory predicts the trends of ecological conditions in which sexual/asexual species of the same taxonomic group (or sexual/asexual stages of the same species) will prevail. Predictions of the “classic” theory with the above-mentioned specifications are compared with predictions of other hypotheses and data from natural observation: only the “classic” theory is confirmed by empirical evidence.

Introduction

The evolutionary justification of gene recombination between two individuals, defined with the technical term “mixis” but usually referred to using the popular word “sex”, is a widely discussed topic (Ghiselin, 1974; Williams, 1975; Maynard Smith, 1978; Bell, 1982; Ridley, 1993).The “classic” hypothesis (alias Fisher-Muller hypothesis) that sexual reproduction is evolutionarily advantageous because it allows a continuous rearrangement of genes (Fig. 1), which Bell called “The Vicar of Bray” (Bell, 1982), was first expressed by Weismann (Weismann, 1889) and later by Guenther (Guenther, 1906). Afterwards, it was been formulated in terms of population genetics by Fisher (Fisher, 1930) and Muller (Muller, 1932) and later, with greater mathematical formalism, by Muller (Muller, 1958, 1964) and Crow and Kimura (Crow and Kimura 1965).Maynard Smith (Maynard Smith, 1968) criticised the “classic” hypothesis with the following, simple but effective, argument.If, in an infinite population of a haploid species, there are two genes (a, b), with alleles (A, B) having an advantage (sA, sB) over a and b, respectively, combinations frequencies in the next generation will be:

Pn+1,ab = Pn,ab / T

Pn+1,Ab = Pn,Ab (1 + sA) / T

Pn+1,aB = Pn,aB (1 + sB) / T

Pn+1,AB = Pn,AB (1 + sAB) / T (1)

where:

Pn,xy = frequency of combination xy at generation n;

Pn+1,xy = frequency of combination xy at generation n+1;

k = interaction (epistasis) between the fitnesses;

T = the sum of numerators;

sAB= [(1 + sA)(1 + sB) - 1] k (2)

If, at generation n, there is no linkage disequilibrium (D), that is, if:

D = Pn,ab Pn,AB - Pn,Ab Pn,aB = 0 (3)

with no epistasis (k = 1), Eq. (1) determine that in the next generation it will be always:

D = Pn+1,ab Pn+1,AB - Pn+1,Ab Pn+1,aB = 0 (4)

with or without recombination, which can only halve linkage disequilibrium at each generation (Maynard Smith, 1978). Therefore, with these conditions sex is not advantageous.With negative linkage disequilibrium (D < 0) sex would be advantageous, while with positive linkage disequilibrium sex would be disadvantageous.If there is positive epistasis (k > 1) between the fitnesses, sex is disadvantageous because it breaks the more advantageous combination AB. The contrary happens if there is negative epistasis (k < 1).Maynard Smith tried to overcome his argument (Maynard Smith, 1978), observing, in particular, that it was valid only for infinite populations but that “linkage disequilibrium is bound to arise by chance in a finite population” (p. 15) and in conditions of negative linkage disequilibrium sex would be advantageous, as previously observed by Felsenstein (Felsenstein, 1974).Many scholars did not accept the counter-arguments of Maynard Smith (Crow and Kimura, 1969; Williams, 1975), and it must be asked why conditions of negative linkage equilibrium, favourable for sex, should prevail over positive occurrences?The doubts about the validity of Fisher-Muller “classic” explanation of sex caused the flourishing of alternative hypotheses such as, to use the eponyms of Bell (Bell, 1982):

- Best-Man (Williams, 1966; Emlen, 1973; Treisman, 1976) (Recombination produces “a few individuals of extraordinarily high fitness. If only these individuals have any appreciable chance of surviving, then sexual parents will contribute a disproportionately large number of progeny to the next generation ...” (Bell, 1982));

- Hitch-hiker (Hill and Robertson, 1966; Felsenstein, 1974) (Stochastically generated linkage disequilibria increase the variance of fitness of any single-locus genotype and so retard the fixation of a favourable allele. An allele increasing the rate of recombination reduces linkage disequilibria and accelerates the fixation of favourable alleles and thus, for selection, it is hitch-hiked by these favourable alleles);

- Tangled Bank (Ghiselin, 1974; Burt and Bell, 1987; Ridley, 1993) (Sex diversifies progeny and its advantage is greater in conditions of environmental spatial heterogeneity, that is various “ecological niches in the same small geographical area – in an environment which does not change in time” (Bell, 1982));

- a plurality of theories is necessary to explain the existence of sex (West et al., 1999);

and others theories, on the whole classified by Kondrashov (Kondrashov 1993).

This paper originates both from the facts that many of these hypotheses are weakened by old serious criticisms (Bell, 1982) and that various subsequent attempts to explain sex advantage in finite populations appear too complex (Kondrashov and Yampolsky, 1996; Bürger, 1999; Pálsson, 2002; Iles et al., 2003; Barton and Otto, 2005; Martin et al., 2006; Tannenbaum, 2008), as well as from the conviction that sex evolutionary advantage must be investigated without hypothesizing artful and / or unduly limiting mechanisms.

I want to formulate a model that shows for sex - in terms of individual selection, as indicated by Felsenstein (Felsenstein, 1974) - both advantage in finite populations and no advantage in infinite populations. Moreover, the model must consider the important suggestion that natural populations are subject to genetic drift and are spatially structured (Otto and Lenormand, 2002).

In the first section, based on the classic Fisher-Muller hypothesis, stated in terms of individual selection and, for the sake of brevity, referred to as the “classic” hypothesis, I will illustrate a model for an infinite population that confirms Maynard Smith’s predictions (Maynard Smith, 1968, 1978).

In the subsequent section, I will insert in the model the condition of a finite population that demonstrates in this case an advantage for sexual reproduction, in accordance with a key observation on Fisher-Muller’s hypothesis expressed by Felsenstein (Felsenstein, 1974): “ ... those authors who have allowed finite-population effects into their models have been the ones who found an advantage to having recombination, while those whose models were completely deterministic found no consistent advantage.” (p. 738)

The method utilized is the precise definition of a theoretical model and the following computer-aided verification, as discussed by Bell (Bell, 1982, pp. 79-84).

Finally, predictions of the “classic” hypothesis are compared with predictions of other theories and with data from natural observation.

Review

The simulation model for infinite populationsLet us consider a species:a) that is haploid;b) with an infinite population;c) with half of the individuals at generation zero having - in a specific locus - a gene R+ allowing conjugation and free recombination only with other individuals having R+, while the others have an allele R- allowing conjugation and recombination only in a fraction z of individuals.If z > 0 the pool of recombining individuals is constituted by all R+ individuals and a fraction z of R- individuals. If z = 0, as in most of the following simulations, there is “no sharp distinction between individual selection and group selection”, as underlined by Felsenstein and Yokoyama (Felsenstein and Yokoyama ,1976), but the selection will actually be considered only in strict terms of individual selection.d) with mutation rates of R+ in R-, namely turning a sexual individual into an asexual individual, or vice versa, of zero frequency;e) with R+ and R- individuals having the same ecological niche and being by no means distinguishable except for the condition expressed in d;f) with the disadvantage for sexual individuals of finding a mate and of coupling and with any other possible disadvantage of sex, the so-called “cost of sex” included, considered negligible;g) with new alleles (A, B, C, ...) more advantageous than those prevailing in the species (a, b, c, ...), supposed at generation zero with frequency = 1;h) with independent gene transmission of any allele", i.e. the recombination fraction is assumed 0.5;i) with the mutation rate, at each generation, of an allele x in X equal to ux and the back-mutation rate of Xin x equal to wx.The question is, whether there is an advantage of sexual on asexual individuals, or vice versa, that is, whether there is a spreading or a decay of R+.The model is restricted to the cases of:I) two genes (a, b) and their respective new alleles (A, B) (“two genes case”), with four possible combinations (ab, Ab, aB, AB);II) three genes (a, b ,c) and the respective new alleles (A, B, C) (“three genes case”), with eight possible combinations (abc, Abc, aBc, abC, ABc, aBC, AbC, ABC).These restrictions are not a limitation, because if sex will be proved advantageous with only 2 or 3 genes, its greater fitness will be self-evident with more genes.

For the sake of simplicity, the following is hypothesized:

u = ua = ub = uc (5)

w = wa = wb = wc (6)

s = sA = sB = sC (7)

With two and three genes, the possible cases of mutations from one combination into another are 10 (Fig. 2 and Table 1) and 38 (Fig. 3 and Table 2), respectively. The probabilities of transformations are indicated in the tables.The fitness for individuals with two advantageous alleles (FXY; XY means AB, for the two genes case; AB or BC or AC, for the three genes case) is:

FXY = 1 + k [(1 + s)2 –1] (8)

(where k = 1 when there is no interaction - or epistasis - between the genes).

In the case of three advantageous alleles:

FABC = 1 + k2 [(1 + s)3 –1] (9)

If we indicate the frequency of combination xy in R+ individuals at the n-th generation with Pxy,n and that in R- individuals with Pxy’,n, recombination for R+ individuals is simulated, in the two genes case, by calculating the frequencies of a, A, b, B, over the total of individuals with R+ (PR+):

Pa,n = Pab,n + PaB,n; ... (10)

and, afterwards, by using the equations:

Pab,n+1 = ½ Pab,n + ½ Pa,n Pb,n PR+,n; ... (11)

PR+,n PR+,n

The first part of the solution of each equation means that, in the recombination between individual I and another individual, in half of the cases the allele present in I does not change. The second part means that, in the remaining 50%, the allele present in I is substituted by other alleles from other individuals: the frequencies of the substituting alleles are given by the multiplication of the relative frequencies of each allele (Px,n / PR+,n), with the result multiplied for the frequency of R+ (PR+,n).On the contrary, for R- individuals and with z = 0 (see condition [c]), there is no calculation:

Pab’,n+1 = Pab’,n ; ... (12)

In the three genes case, recombination for R+ individuals is simulated by calculating the frequencies of a, A, b, B, c, C over the total of individuals with R+:

Discussion

A comparative and experimental critique of the theories(Empirical evidence for various theories)As a theory is sound or unsound according to whether predictions are confirmed or falsified by empirical data, it is necessary to verify whether predictions of the “classic” hypothesis about sex evolutionary advantage are confirmed or refuted by data from natural observation. Moreover, according to the scientific method, a theory refuted by empirical data must be considered as untenable and not presented as a valid hypothesis until its contradictions with empirical data have been explained or somehow resolved.I have drawn a table (Table 3) in which predictions of the “classic” hypothesis on the evolutionary meaning of sex, along with those of three other theories (Best-Man, Tangled Bank and Red Queen) concerning the expected trends of the distribution in the nature of sex and related phenomena are compared with empirical evidence from natural observation.This paragraph has the same name as chapter 4 of Bell’s book (Bell, 1982) and has its aims, methods and predictions for the Best-Man, Tangled-Bank and Red Queen hypotheses and references to data from natural observation, in common with it. Predictions for the aforesaid hypotheses are identical to those expounded by Bell, but in some cases, in the absence of Bell’s predictions, I have attempted a prediction explained in an appropriate note.I have also formulated predictions of the “classic” hypothesis with one simple criterion: as the theory and the simulation model of this paper maintain and show that sex – disregarding DS - is always advantageous except in small and isolated populations, sex is predicted to be always favoured except for the above-mentioned populations and when DS is important (severely disturbed environments, r-selection, phases of exponential growth of population, etc.). Moreover, because DS does not exist as regards recombination, no correlation between certain phenomena of recombination (achiasmy, chromosome number, crossing over frequency) and amphimixis or parthenogenesis is expected.In various cases, predictions of the “classic” hypothesis and those of other hypotheses coincide but the motivations are different (e.g., predictions of the “classic” hypothesis and those of the Red Queen for Correlation with different habitats).The noteworthy result, in my judgement, is an almost total correspondence between predictions of the “classic” hypothesis and data from natural observation.The utter failure of the Best-Man hypothesis is remarkable and I share Bell’s negative opinion on this theory which, in Table 3, has the only function of showing a plain example of a hypothesis in almost constant contradiction with data from natural observation.For the Tangled-Bank and the Red Queen hypotheses, there are the wrong predictions of correlation between certain phenomena of recombination and amphimixis/parthenogenesis (“Amphimixis is to parthenogenesis as high rates of recombination are to low; the correlates of low levels of recombination will therefore be the same as the correlates of parthenogenesis.” (Bell, 1982)), a significant contradiction described and underlined by Bell in ch. 5.2 (Bell 1982). On the other hand, as for such phenomena as achiasmy, frequency of crossing over and number of chromosomes intrinsically DS does not exist, a correlation between these phenomena and parthenogenesis is not predicted by the “classic” hypothesis, in accordance with data from natural observation (Bell, 1982).Moreover, for the Tangled-Bank hypothesis the prediction for parasitism is not completely adequate, as the Bell, himself, underlines (Bell, 1982).As regards other theories not considered in the table:- Muller’s Ratchet hypothesis. This could justify sex only for small populations as “Muller’s ratchet operates only in small or asexual populations ... harmful mutations are unlikely to become fixed in sexual populations unless the effective population size is very small.” (Keightley and Otto, 2006). Therefore, this theory, which is not contradicted by the results of this paper, could integrate the “classic” theory.- Historical hypothesis. This theory, which does not justify sex existence, is refuted by the evidence that sexual or asexual reproduction is influenced by many conditions. However, if it is considered not as a theory explaining sex but as an inertial factor restraining a free passage from sexual to asexual reproduction, or vice versa, it should deserve a certain amount of attention.- Hitch-hiker hypothesis. A R+ gene could be described as a gene that is advantaged because it hitchhikes favourable genes that are better spread because of by its action. The hitch-hiker hypothesis could, therefore, be defined as a different and indirect way of expounding the “classic” theory.- Hypothesis that sex is advantageous because it slows down evolution and excessive specialisation. This theory makes no prediction.

Conclusion(s)

Williams proclaimed (Williams, 1975): “... the unlikelihood of anyone ever finding a sufficiently powerful advantage in sexual reproduction with broadly applicable models that use only such general properties as mutation rates, population sizes, selection coefficients, etc.” (p. 14), and Ridley wrote (Ridley, 1993): “I asked John Maynard Smith, one of the first people to pose the question ‘Why sex?’, whether he still thought some new explanation was needed. ‘No. We have the answers. We cannot agree on them, that is all.’ ” (p. 29).I think that Williams’ unlikelihood is now a likelihood and that the uncertainty of Maynard Smith has been solved: with theoretical arguments, the advantage of sex has been rationally explained by the “classic” theory in terms of individual selection and using only the “general properties ...” that Williams insisted on (Williams, 1975). Moreover, if we consider the disadvantage of sex, it is possible to formulate predictions about the trends of its diffusion in nature that are confirmed by data from natural observation.For small populations, Muller’s Ratchet hypothesis, if confirmed, could reinforce and integrate the “classic” theory. Historical hypothesis deserves attention as an inertial factor in the prediction of trends of diffusion of sex and related phenomena.The correct concept that biotic factors – often with oscillating s values - are quantitatively more important than physical factors as selective forces in determining evolution (Red Queen concept), which is the pivotal idea at the roots of the Red Queen theory, is not at all against the “classic” theory, although it is insufficient in itself to explain sex, and should be considered an argument that reinforces this hypothesis.Somehow, the “pluralist approach to sex and recombination” (West et al., 1999) seem to be the correct solution, but with this specification: “classic” theory is the trunk with the main branches and other theories complete the tree.

if the number of iterations is not small, the graphic display of simulations may disappear after some simulations (as a consequence of PC power limits) and program commands freeze. This does not mean that the program is blocked: it continues to run till the end. Please, await the end and then see the results in ReportFile2.txt (result of each simulation) and ReportFile3.txt (mean and SD for each group of simulation).

Source(s) of Funding

none

Competing Interests

none

Disclaimer

This article has been downloaded from WebmedCentral. With our unique author driven post publication peer
review, contents posted on this web portal do not undergo any prepublication peer or editorial review. It is
completely the responsibility of the authors to ensure not only scientific and ethical standards of the manuscript
but also its grammatical accuracy. Authors must ensure that they obtain all the necessary permissions before
submitting any information that requires obtaining a consent or approval from a third party. Authors should also
ensure not to submit any information which they do not have the copyright of or of which they have transferred
the copyrights to a third party.
Contents on WebmedCentral are purely for biomedical researchers and scientists. They are not meant to cater to
the needs of an individual patient. The web portal or any content(s) therein is neither designed to support, nor
replace, the relationship that exists between a patient/site visitor and his/her physician. Your use of the
WebmedCentral site and its contents is entirely at your own risk. We do not take any responsibility for any harm
that you may suffer or inflict on a third person by following the contents of this website.

What is article Popularity?

Article popularity is calculated by considering the scores: age of the articlePopularity = (P - 1) / (T + 2)^1.5
WhereP : points is the sum of individual scores, which includes article Views, Downloads, Reviews, Comments and their weightage

Scores

Weightage

Views Points

X

1

Download Points

X

2

Comment Points

X

5

Review Points

X

10

Points= sum(Views Points + Download Points + Comment Points + Review Points) T : time since submission in hours.
P is subtracted by 1 to negate submitter's vote.
Age factor is (time since submission in hours plus two) to the power of 1.5.factor.

How Article Quality Works?

For each article Authors/Readers, Reviewers and WMC Editors can review/rate the articles. These ratings are used to determine Feedback Scores.

In most cases, article receive ratings in the range of 0 to 10. We calculate average of all the ratings and consider it as article quality.